\(\int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^4} \, dx\) [1265]

Optimal result

Integrand size = 22, antiderivative size = 49 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^4} \, dx=\frac {100 x}{81}-\frac {49}{729 (2+3 x)^3}+\frac {259}{243 (2+3 x)^2}-\frac {503}{81 (2+3 x)}-\frac {740}{243} \log (2+3 x) \]

[Out]

100/81*x-49/729/(2+3*x)^3+259/243/(2+3*x)^2-503/81/(2+3*x)-740/243*ln(2+3*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^4} \, dx=\frac {100 x}{81}-\frac {503}{81 (3 x+2)}+\frac {259}{243 (3 x+2)^2}-\frac {49}{729 (3 x+2)^3}-\frac {740}{243} \log (3 x+2) \]

[In]

Int[((1 - 2*x)^2*(3 + 5*x)^2)/(2 + 3*x)^4,x]

[Out]

(100*x)/81 - 49/(729*(2 + 3*x)^3) + 259/(243*(2 + 3*x)^2) - 503/(81*(2 + 3*x)) - (740*Log[2 + 3*x])/243

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {100}{81}+\frac {49}{81 (2+3 x)^4}-\frac {518}{81 (2+3 x)^3}+\frac {503}{27 (2+3 x)^2}-\frac {740}{81 (2+3 x)}\right ) \, dx \\ & = \frac {100 x}{81}-\frac {49}{729 (2+3 x)^3}+\frac {259}{243 (2+3 x)^2}-\frac {503}{81 (2+3 x)}-\frac {740}{243} \log (2+3 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^4} \, dx=\frac {-11803-23193 x+24057 x^2+64800 x^3+24300 x^4-2220 (2+3 x)^3 \log (2+3 x)}{729 (2+3 x)^3} \]

[In]

Integrate[((1 - 2*x)^2*(3 + 5*x)^2)/(2 + 3*x)^4,x]

[Out]

(-11803 - 23193*x + 24057*x^2 + 64800*x^3 + 24300*x^4 - 2220*(2 + 3*x)^3*Log[2 + 3*x])/(729*(2 + 3*x)^3)

Maple [A] (verified)

Time = 2.29 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.65

method	result	size
risch	\(\frac {100 x}{81}+\frac {-\frac {503}{9} x^{2}-\frac {5777}{81} x -\frac {16603}{729}}{\left (2+3 x \right )^{3}}-\frac {740 \ln \left (2+3 x \right )}{243}\)	\(32\)
norman	\(\frac {\frac {6649}{162} x +\frac {15367}{108} x^{2}+\frac {31003}{216} x^{3}+\frac {100}{3} x^{4}}{\left (2+3 x \right )^{3}}-\frac {740 \ln \left (2+3 x \right )}{243}\)	\(37\)
default	\(\frac {100 x}{81}-\frac {49}{729 \left (2+3 x \right )^{3}}+\frac {259}{243 \left (2+3 x \right )^{2}}-\frac {503}{81 \left (2+3 x \right )}-\frac {740 \ln \left (2+3 x \right )}{243}\)	\(40\)
parallelrisch	\(-\frac {159840 \ln \left (\frac {2}{3}+x \right ) x^{3}-64800 x^{4}+319680 \ln \left (\frac {2}{3}+x \right ) x^{2}-279027 x^{3}+213120 \ln \left (\frac {2}{3}+x \right ) x -276606 x^{2}+47360 \ln \left (\frac {2}{3}+x \right )-79788 x}{1944 \left (2+3 x \right )^{3}}\)	\(60\)
meijerg	\(\frac {3 x \left (\frac {9}{4} x^{2}+\frac {9}{2} x +3\right )}{16 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {x^{2} \left (3+\frac {3 x}{2}\right )}{16 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {59 x^{3}}{48 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {5 x \left (\frac {99}{2} x^{2}+45 x +12\right )}{162 \left (1+\frac {3 x}{2}\right )^{3}}-\frac {740 \ln \left (1+\frac {3 x}{2}\right )}{243}+\frac {20 x \left (\frac {405}{8} x^{3}+\frac {495}{2} x^{2}+225 x +60\right )}{243 \left (1+\frac {3 x}{2}\right )^{3}}\)	\(104\)

[In]

int((1-2*x)^2*(3+5*x)^2/(2+3*x)^4,x,method=_RETURNVERBOSE)

[Out]

100/81*x+27*(-503/243*x^2-5777/2187*x-16603/19683)/(2+3*x)^3-740/243*ln(2+3*x)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.27 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^4} \, dx=\frac {24300 \, x^{4} + 48600 \, x^{3} - 8343 \, x^{2} - 2220 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) - 44793 \, x - 16603}{729 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]

[In]

integrate((1-2*x)^2*(3+5*x)^2/(2+3*x)^4,x, algorithm="fricas")

[Out]

1/729*(24300*x^4 + 48600*x^3 - 8343*x^2 - 2220*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) - 44793*x - 16603)/(2
7*x^3 + 54*x^2 + 36*x + 8)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^4} \, dx=\frac {100 x}{81} + \frac {- 40743 x^{2} - 51993 x - 16603}{19683 x^{3} + 39366 x^{2} + 26244 x + 5832} - \frac {740 \log {\left (3 x + 2 \right )}}{243} \]

[In]

integrate((1-2*x)**2*(3+5*x)**2/(2+3*x)**4,x)

[Out]

100*x/81 + (-40743*x**2 - 51993*x - 16603)/(19683*x**3 + 39366*x**2 + 26244*x + 5832) - 740*log(3*x + 2)/243

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^4} \, dx=\frac {100}{81} \, x - \frac {40743 \, x^{2} + 51993 \, x + 16603}{729 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac {740}{243} \, \log \left (3 \, x + 2\right ) \]

[In]

integrate((1-2*x)^2*(3+5*x)^2/(2+3*x)^4,x, algorithm="maxima")

[Out]

100/81*x - 1/729*(40743*x^2 + 51993*x + 16603)/(27*x^3 + 54*x^2 + 36*x + 8) - 740/243*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^4} \, dx=\frac {100}{81} \, x - \frac {40743 \, x^{2} + 51993 \, x + 16603}{729 \, {\left (3 \, x + 2\right )}^{3}} - \frac {740}{243} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

integrate((1-2*x)^2*(3+5*x)^2/(2+3*x)^4,x, algorithm="giac")

[Out]

100/81*x - 1/729*(40743*x^2 + 51993*x + 16603)/(3*x + 2)^3 - 740/243*log(abs(3*x + 2))

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^2 (3+5 x)^2}{(2+3 x)^4} \, dx=\frac {100\,x}{81}-\frac {740\,\ln \left (x+\frac {2}{3}\right )}{243}-\frac {\frac {503\,x^2}{243}+\frac {5777\,x}{2187}+\frac {16603}{19683}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}} \]

[In]

int(((2*x - 1)^2*(5*x + 3)^2)/(3*x + 2)^4,x)

[Out]

(100*x)/81 - (740*log(x + 2/3))/243 - ((5777*x)/2187 + (503*x^2)/243 + 16603/19683)/((4*x)/3 + 2*x^2 + x^3 + 8
/27)